Assembly of planar chiral superlattices from achiral building blocks

The spontaneous assembly of chiral structures from building blocks that lack chirality is fundamentally important for colloidal chemistry and has implications for the formation of advanced optical materials. Here, we find that purified achiral gold tetrahedron-shaped nanoparticles assemble into two-dimensional superlattices that exhibit planar chirality under a balance of repulsive electrostatic and attractive van der Waals and depletion forces. A model accounting for these interactions shows that the growth of planar structures is kinetically preferred over similar three-dimensional products, explaining their selective formation. Exploration and mapping of different packing symmetries demonstrates that the hexagonal chiral phase forms exclusively because of geometric constraints imposed by the presence of constituent tetrahedra with sharp tips. A formation mechanism is proposed in which the chiral phase nucleates from within a related 2D achiral phase by clockwise or counterclockwise rotation of tetrahedra about their central axis. These results lay the scientific foundation for the high-throughput assembly of planar chiral metamaterials.

Assembly of Au Td@Ag cube NPs for purification: Self-assembly has been widely used to purify NPs with large surface area since they can form extended assemblies and precipitate (e.g., cubes, prisms, high aspect ratio nanorods, etc) [1][2][3] . Since the target Au Td NPs could be selectively embedded in Ag cubes via overgrowth, we hypothesized they could be purified by considering the assembly of cube-shaped particles. To determine the purification conditions, especially the concentration of surfactant, we used the following empirical formula for interparticle attraction that has been used in previous reports to predict selective assembly and precipitation [1][2][3] : where kBT is the thermal energy, rm the micelle radius, A the particle-particle facet contact area, N0 Avogadro's number, c the concentration of surfactant, cmc the critical micelle concentration, and n the number of molecules that assemble to form a micelle. For CTAC solutions, the rm, cmc and n are 3 nm, 1 mM and 120, respectively. For a typical Au Td@Ag cube NP, the cube edge length is 70 nm, leading to a possible contact area, A, of 4900 nm 2 . According to previous reports, when the empirical potential |U| reaches 4~5kBT, the targeted NPs will assemble and precipitate, suggesting that a CTAC concentration around 28 -35 mM will result in Au Td@Ag cube NP assembly. It is important to mention that due to the batch-to-batch differences, it is necessary to measure the size of Au Td@Ag cube NPs before purification and adjust conditions accordingly; most of our samples required 32 mM CTAC to purify Au Td@Ag cube NPs with 70 nm edge length. Multiple rounds of precipitation and separation with this method lead to increasingly pure samples.

Tetrahedron structure analysis and calculations:
To have a complete understanding of the formation of the chiral hexagonal (3) phase, we considered a series of related 2D structures with different packing of Td particles. Due to the nature of the interactions between particles in our system, assemblies that have the highest face-to-face contact area and smallest interparticle distance will be those of minimum energy, i.e., most favorable. Since we are only considering structures capable of forming extended 2D crystals, we exclude from our analysis Td assemblies that result in local decahedral or icosahedral packing and lack long-range order.
2.a. For achiral (1) assemblies: we consider a structure where a tetrahedron dimer with hexagonshaped face-to-face contact area is arrayed in a 2D hexagonal Bravais lattice (Supplementary Figure 1). The extended material resembles two interpenetrated sheets of Td, each with in-plane triangular tip-totip packing (Supplementary Figure 1A, 1B). This results in a hexagonal contact area (Supplementary Figure 1C, 1D) and interparticle distance, d (Supplementary Figure 1E).
For tetrahedra with edge length (L), the overlap area (S) in Supplementary Figure 1D is given by a hexagon with edge length of L/3: The minimum interparticle distance (dmin) can be obtained with the model in Supplementary Figure 1B using the equations of two planes. We consider a cube with edge length a, the diagonal of which produces a Td with edge length L. Two typical planes as shown in Supplementary Figure 1C and 1E can be calculated by two sets of points: Plane I: Plane II: The corresponding equations for above planes are: Plane I: x + y + z = 2a Plane II: x + y + z = 2.5a The distance (d) between two parallel planes is: Since the edge length (L) of tetrahedra is: and since this is the closest-packed Td can get in the achiral (1) phase given their tip sharpness, the minimum distance (dmin) between two tetrahedra NPs is: 2.b. For achiral (2) assemblies: we considered a structure identical to achiral (1) in which the top layer of Td (tips pointing down, Supplementary Figure 2A below) are shifted along a unit vector such that each particle has one nearest neighbor (NN) with face-to-face spacing smaller than the dmin calculated above but two next-nearest neighbors (NNN) with spacing larger than dmin. Since the achiral (1) structure is unfavorable because of the geometric constraint imposed by the sharp tips that sets a large value for dmin, we hypothesized that this translation of the top layer of particles might result in a lower energy configuration. This structure maintains the tip-to-tip packing of achiral (1) with a hexagonshaped NN contact area and a smaller elongated hexagonal parallelogram-shaped NNN contact area.
After shifting a layer of tetrahedra to a new NN particle distance of d* (Supplementary Figure 2A, red arrow) a parameter Δd can be defined as: The NNN interparticle distance (d') can be regarded as the original distance between two tetrahedra plus the movement distance: 2.c. For the packing of tetrahedra with rounded tips: Chemically etching tetrahedra results in selective rounding of their tips (Supplementary Figure 3A, below), which results in a decreased minimum interparticle distance (Supplementary Figure 3B) while maintaining a large contacting area. We define the tip radius of curvature (R, Supplementary Figure 3C) and calculate the resulting minimum the interparticle distance (d min R ) using (H), the projection of the particle-particle movement. Based on the geometrical model (Supplementary Figure 3C inset, 3D), H can be calculated from: (Supplementary eq. 13) H can be obtained: (Supplementary eq. 14) The minimum interparticle distance as a function of tip rounding (d min R ) is:  (Fig. S2) which resolves into a distinct peak located at ~541 nm after separation, demonstrating the improved purity.
Supplementary Figure 11. Analysis of the purity of Au Td NPs from TEM images. Au Td NPs were first deposited on TEM grids by fast evaporation to make them disperse evenly, after which a series of (a) TEM images were taken from different areas to sample a representative population. (b) Red and blue dots were assigned to particles based on whether they belong to Td or impurity categories, respectively. More than 17,000 particles were counted by this method to determine the purity of separated Au Td samples. All scale bars are all 200 nm.

a b
Supplementary  Figure 13. Setup for the self-assembly of tetrahedra on a substrate. a) Au Td NPs assembled on substrates in high humidity allows for control of the evaporation rate, inset is a photo of the experimental setup; b) Schematic illustration of the assembly process of Au Td NPs during evaporation. Over time, the concentration of Au Td NPs, electrolytes, and micelles increases, creating a balance of forces which drives 2D assembly on the substrate. The position of the local minimum of this function indicates the equilibrium particle spacing (d*) and interaction strength (Utot*). This calculation is then repeated for a range of increasing CTAC concentrations, which generally causes particle faces to pack more closely (b) and with stronger (c, more negative Utot*) attraction. While plots of d* vs. CTAC show very little dependence on the superlattice configuration, Utot*vs. CTAC is highly dependent on the facet overlap area experienced between neighboring Td in an assembly and thus can be used to identify the phase behavior of the system; more negative values of Utot* at a given CTAC concentration indicate a more stable superlattice. Of all the forces involved in Td colloidal interactions, vdW is the only one dependent on a volume element, as opposed to a surface element, and therefore is the only one that will change as a function of particle thickness. Since the contribution of the van der Waals attraction to the overall interaction energy is accurately captured by considering only the first 5-10 nm of particle depth, we conclude that approximating the total particle interaction as a sum of 4 independent Td facet interactions with infinite depth is a reasonable simplifying assumption that captures the relevant system behavior (see methods section).
Supplementary Figure 18. Breakdown of the different contributions to the overall interaction potential involved in the assembly of Au Td NPs as function of CTAC concentration shown as absolute values (a) and percentage of total (b). van der Waals interactions dominate at low CTAC concentration while depletion attraction dominates at high CTAC concentration.
Supplementary Figure 19. Comparison of the facet overlap areas for a single Td particle in a (a) 2D and (b) 3D hexagonal chiral superlattice. For the 2D phase, each particle has three elongated hexagonal parallelogramshaped overlaps while for 3D lattice, each particle has three elongated hexagonal parallelogram-shaped overlaps and one triangular overlap. These overlap areas are used to calculate and compare Utot * for 2D vs. 3D superlattices. interactions, whose length scale is set by the size of the CTAC micelle ~5 -10 nm, become significant and can compete with vdW and electrostatic forces. The result is that the net interaction adopts a local minimum value, such that Utot* is well-defined (a), and Td rise off the surface with d* ≈ 9 nm (b). We propose that the range of CTAC concentrations for which Td are strongly-bound to the substrate (0 -100 mM) is a "nucleation regime" where particles are drawn out of solution and concentrated at substrates, which facilitates the crystallization of 2D superlattice phases. If depletion forces are weakened to 10% of their original value (red data), which is consistent with rough surfaces, then the CTAC concentration at which Td rise off the substrate is lowered. This means that less time is spent in the nucleation regime during droplet evaporation and interparticle interactions are weaker during that time, both of which suggest fewer superlattice nuclei and/or more disordered assemblies. Also note that for rough substrates, when Td rise off the surface, Utot* is only ~6 kT, which may too weak to hold particles to the interface for long enough to allow 2D assembly to occur. Thus, substrates appear to act as nucleation sites for Td superlattices which, in addition to their kinetically preferred growth behavior, strongly favors 2D crystals. The model predicts that Td with tip radii smaller than 7.5 nm should assemble into chiral superlattices but above 7.5 nm should assemble into achiral superlattices. Although our ability to control tip radius of curvature is limited to ~ 1 nm, we experimentally observe that Td with tip radii of 7.5 nm assemble into chiral superlattices but 8.4 nm assemble into achiral superlattices (Figure 3f), showing excellent agreement with the predictions of the phase diagram.  Figure 31k). This is a consequence of the Td size-dependence of dmin, which is the steric threshold driving the transition from achiral to chiral superlattices via rotation (see maintext). All scale bars are 500 nm.